Let \overset{\rightarrow}{a} , \textrm{ }\textrm{ } \overset{\rightarrow}{b} \textrm{ }\textrm{ } and \textrm{ }\textrm{ }\textrm{ } \overset{\rightarrow}{c}be three non-zero vectors such that no two of them are collinear and\left(\right. \overset{\rightarrow}{a} \times \overset{\rightarrow}{b} \left.\right) \times \overset{\rightarrow}{c} = \frac{1}{3} \left|\right. \textrm{ } \overset{\rightarrow}{b} \textrm{ } \left|\right. \textrm{ }\textrm{ } \left|\right. \textrm{ } \overset{\rightarrow}{c} \textrm{ } \left|\right. \textrm{ }\textrm{ } \overset{\rightarrow}{a}. If \theta is the angle between vectors \overset{\rightarrow}{b} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{c}, then a value of sin \theta is

Engineering

Mathematics

Vector Triple Product

Question

Let $\vec{a}, \vec{b} and \vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3} | \vec{b} | | \vec{c} | \vec{a}$ . If $θ$ is the angle between vectors $\vec{b} and \vec{c}$ , then a value of sin $θ$ is

$\frac{- \sqrt{2}}{3}$

$\frac{2}{3}$

$\frac{- 2 \sqrt{3}}{3}$

$\frac{2 \sqrt{2}}{3}$

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$(\vec{a} . \vec{c}) \vec{b} - (\vec{b} . \vec{c}) \vec{a} = \frac{1}{3} | \vec{b} | | \vec{c} | \vec{a}$

$(\vec{a} . \vec{c}) \vec{b} = (\vec{b} . \vec{c} + \frac{1}{3} | \vec{b} | | \vec{c} |) \vec{a}$

$\vec{b} . \vec{c} + \frac{1}{3} | \vec{b} | | \vec{c} | = 0$ $\Rightarrow cosθ = - \frac{1}{3} \Rightarrow \sin θ = \frac{2 \sqrt{2}}{3}$